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Well-defined Deligne-to-ribbon functor and geodesic images

PDF p. 23: Lemma 6.1 and proof

Statement

Let $Γ$ be a finite-rank spherical Coxeter diagram, so its Coxeter group $W$ is finite, and fix a subset $J$ of its vertices. Use the chamber labels $(x, I)$ , positive arrangement category $D^{+} (J)$ , its groupoid completion $D (J)$ , and reduced ribbon groupoid $R (J)$ of the indicated definitions. A geodesic is a chamber path of minimum crossing length among paths with the same endpoints. For $w \in W$ , write $σ_{w}$ for its positive reduced-word lift in the Artin group.

The assignment $G (x, I) = I$ and $G (ω_{a, I}) = μ_{a, I}$ extends to a well-defined functor $G : D (J) \to R (J)$ . If a geodesic travels through labels $(x_{1}, I_{1}), \dots, (x_{n}, I_{n})$ , with crossing indices $a_{i}$ , its image is the arrow $I_{1} \to I_{n}$ labelled by $σ_{x_{1}^{- 1} x_{n}} = μ_{a_{1}, I_{1}} \dots μ_{a_{n - 1}, I_{n - 1}}$ . In particular its image depends only on its endpoints, including when the path is empty.

Statement references

Assignment from wall crossings to ribbon generators

Positive arrangement category and its groupoid completion

Reduced ribbon groupoid

Simple wall-crossing elements and maps

Reduced-word lifts and the prefix-order convention

Facts & Assumptions

[F1]

The proposed object and crossing assignments form a directed graph map into $R (J)$ , preserving sources and targets. Its crossing at $(x, I)$ with index $a$ has target first coordinate $x v_{a, I}$ . Assignment from wall crossings to ribbon generators Simple wall-crossing elements and maps

[F2]

The positive category is the chamber-path category modulo the congruence generated by identifying pairs of geodesic paths with common endpoints. Its groupoid completion is $D (J)$ . Positive arrangement category and its groupoid completion

[F3]

For a geodesic with successive wall elements $v_{i} = v_{a_{i}, I_{i}}$ , one has $ℓ (v_{1} \dots v_{n - 1}) = \sum_{i} ℓ (v_{i})$ , including the empty product. Geodesics are exactly length-additive wall words

[F4]

Each elementary ribbon $μ_{a, I}$ is the positive lift $σ_{v_{a, I}}$ . Positive lifts are independent of reduced words and multiply across a length-additive product. Ribbon generators lift wall-crossing morphisms Reduced-word lifts and the prefix-order convention

[F5]

Arrows of $R (J)$ are arrows of the ambient ribbon groupoid tagged by source and target and labelled by Artin-group elements; composition multiplies the labels. The target is a groupoid. Reduced ribbon groupoid

[F6]

Every functor from a small category to a groupoid factors uniquely through its groupoid completion. Groupoid completion and its universal property

Proof

Given: The spherical Coxeter data and fixed subset $J$ in the Statement. Composition of arrows is read from left to right, and $ℓ$ is Coxeter word length.

Proof technique: direct.

Goal: Prove the geodesic image formula and show that the graph assignment extends to the stated functor.

1.1F1F3F4given

Let $p$ be a geodesic with the labels and indices in the Statement, and write $v_{i} = v_{a_{i}, I_{i}}$ . By the crossing formula in [F1], $x_{i + 1} = x_{i} v_{i}$ ; hence $v_{1} \dots v_{n - 1} = x_{1}^{- 1} x_{n}$ . Choose a reduced Coxeter word for each $v_{i}$ . By [F3] their concatenation has length equal to the length of its product, so it is a reduced word for $x_{1}^{- 1} x_{n}$ . By [F4], the product of the corresponding positive lifts is therefore $σ_{x_{1}^{- 1} x_{n}}$ , and each factor is $μ_{a_{i}, I_{i}}$ . For $n = 1$ both products are empty and give the identity.

2.1F1F2F5step 1.1

By [F1], assign to each chamber path the composite of its assigned crossing arrows, with empty paths sent to identities. This respects concatenation. If two geodesic paths have the same endpoint labels, step 1.1 gives them the same Artin-group label; their source and target in $R (J)$ are also the same. They are therefore the same arrow by [F5]. Substitution of equal composites into longer paths shows that the assignment respects the congruence in [F2], so it descends to a functor $D^{+} (J) \to R (J)$ .

3.1F2F5F6step 1.1step 2.1∎

Since $R (J)$ is a groupoid, [F6] extends this functor uniquely through the completion $D (J)$ . Its values on objects and generating crossings are the prescribed ones, and step 1.1 gives the asserted geodesic formula and endpoint independence.

Source notes

The cited article attributes the statement and uses the same geodesic-lift reduction. The proof above uses the declared suppliers and does not assume Lemma 6.1 as a premise.

Depends on

Used by · 2

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 23, Lemma 6.1 and proof